A216078 Number of horizontal and antidiagonal neighbor colorings of the odd squares of an n X 2 array with new integer colors introduced in row major order.

1, 1, 3, 7, 27, 87, 409, 1657, 9089, 43833, 272947, 1515903, 10515147, 65766991, 501178937, 3473600465, 28773452321, 218310229201, 1949230218691, 16035686850327, 153281759047387, 1356791248984295, 13806215066685433, 130660110400259849, 1408621900803060705

Offset

1,3

Comments

Number of vertex covers and independent vertex sets of the n-1 X n-1 white bishops graph. Equivalently, the number of ways to place any number of non-attacking bishops on the white squares of an n-1 X n-1 board. - Andrew Howroyd, May 08 2017

Number of pairs of partitions (A<=B) of [n-1] such that the nontrivial blocks of A are of type {k,n-1-k} if n is even, and of type {k,n-k} if n is odd. - Francesca Aicardi, May 28 2022

Links

R. H. Hardin, Table of n, a(n) for n = 1..210

Eric Weisstein's World of Mathematics, Independent Vertex Set

Eric Weisstein's World of Mathematics, Vertex Cover

Eric Weisstein's World of Mathematics, White Bishop Graph

Formula

a(n) = Sum_{k=0..m} binomial(m, k)*Bell(m+k+e), with m = floor((n-1)/2), e = (n+1) mod 2 and where Bell(n) is the Bell exponential number A000110(n). - Francesca Aicardi, May 28 2022

From Vaclav Kotesovec, Jul 29 2022: (Start)

a(2*k) = A020556(k).

a(2*k+1) = A094577(k). (End)

Example

Some solutions for n = 5:
  x 0   x 0   x 0   x 0   x 0   x 0   x 0   x 0   x 0   x 0
  1 x   1 x   1 x   1 x   1 x   1 x   1 x   1 x   1 x   1 x
  x 2   x 0   x 0   x 2   x 0   x 1   x 1   x 2   x 2   x 1
  0 x   2 x   1 x   3 x   1 x   0 x   2 x   3 x   0 x   0 x
  x 3   x 1   x 2   x 2   x 0   x 1   x 1   x 1   x 2   x 0
There are 4 white squares on a 3 X 3 board. There is 1 way to place no non-attacking bishops, 4 ways to place 1 and 2 ways to place 2 so a(4) = 1 + 4 + 2 = 7. - Andrew Howroyd, Jun 06 2017

Maple

a:= n-> (m-> add(binomial(m, k)*combinat[bell](m+k+e)
           , k=0..m))(iquo(n-1, 2, 'e')):
seq(a(n), n=1..26);  # Alois P. Heinz, Oct 03 2022

Mathematica

a[n_] := Module[{m, e}, {m, e} = QuotientRemainder[n - 1, 2];
   Sum[Binomial[m, k]*BellB[m + k + e], {k, 0, m}]];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Jul 25 2022, after Francesca Aicardi *)

Crossrefs

Column 2 of A216084.

Row sums of A274106(n-1).

Cf. A020556, A094577, A216332, A201862, A286423.

Sequence in context: A148745 A148746 A148747 * A148748 A148749 A098465

Adjacent sequences: A216075 A216076 A216077 * A216079 A216080 A216081

Keyword

nonn

Author

R. H. Hardin, Sep 01 2012

Status

approved